Weight enumerator for second-order Reed-Muller codes

In this paper, we establish the following result. Theorem: Ai, the number of codewords of weight i in the secondorder binary Reed-Muller code of length 2m, is given by Ai = 0 unless i = 2m-1 or 2m-1 f 2m+-i, for some j, 0 < j < [m/2], A0 = A2m =l,and Azmj(f+l) (2” 1>(2+l 1) 1*zsn-,-i = 2 1 4-l I ‘(2m-2 i 1)(2”-3 1) . 42 1 . . . 1 (2m-2i+2 _ 1)(2m-Zi+l _ 1) . 1 4’ 1 I , 1 I j I [M4 INTRODUCTION S SHOWN by Berlekamp ([l], sec. 15.3), the &h-order Reed-Muller (RM) code of length 2” contains 2k codewords, where lc = c:+ (1). If a codeword is written as e = [C,, C,, C,, C,, . . . , C,.b-,], Mansucript received October 6, 1969. The authors are with the Bell Telephone Laboratories, Inc., Murrav Hill, N. J. 07974. ” In the special case r = 2, a theorem of McEliece [5] guarantees that every weight is divisible by 21(m-1)‘21, thus, the only weights between 0 and 2d = 2”‘-l must be of the form 2”-l 2’, [(m 1)/2] 5 i < m 1. Since the code contains the all-one codeword of weight 2”, the only its coefficients may be evaluated by Ci = F(X,,i, -Li, e-e , .L.i), where F(., ., . .. , .) is an m-variate binary polynomial, of degree at most T, and the binary elements X,, i, &,i, *’ , X,,i are obtained from the equation